Schur product with operator-valued entries

TL;DR

This paper characterizes bounded Toeplitz matrices with operator-valued entries and describes Schur multipliers on Hilbert space-valued sequence spaces using operator-valued measures.

Contribution

It provides a new characterization of Toeplitz matrices with operator entries and links Schur multipliers to operator-valued measures.

Findings

Characterization of bounded Toeplitz matrices with operator-valued entries

Description of Schur multipliers via operator-valued measures

Extension of scalar results to operator-valued setting

Abstract

In this paper we characterize Toeplitz matrices with entries in the space of bounded operators on Hilbert spaces $\mathcal{B}(H)$ which define bounded operators acting on $\ell^2(H)$ and use it to get the description of the right Schur multipliers acting on $\ell^2(H)$ in terms of certain operator-valued measures.